Construct and interpret time series plots, identify trend, seasonality, cyclical and irregular variation, smooth a series using moving averages, calculate and apply seasonal indices to deseasonalise data, and fit a trend line to forecast future values

What this dot point is asking

QCAA wants you to handle data collected over time: plot it, name the patterns you see, smooth out the noise, remove the seasonal swing, and then forecast. Time series is the second half of Unit 3 Topic 1. It rewards careful, methodical arithmetic rather than clever insight, which makes it reliable marks in IA2 and the external assessment if you keep your steps tidy.

The answer

The time series plot and its components

A time series plot graphs the variable against time, with consecutive points joined by line segments. You describe the behaviour using four components.

• Trend. The long-term upward or downward movement.
• Seasonality. A pattern that repeats over a fixed, known period (for example every four quarters or every twelve months).
• Cyclical variation. Longer-term rises and falls with no fixed period (for example a business cycle).
• Irregular (random) variation. Unpredictable short-term fluctuation left over once the other components are accounted for.

Smoothing with moving averages

A moving average replaces each value with the average of itself and its neighbours, smoothing out irregular variation to reveal the trend. For an odd number of points (for example a $3$-point or $5$-point moving average) the smoothed value lines up directly with the middle time period.

For an even number of points (a $4$-point moving average for quarterly data) the average falls between two periods, so you apply centring: take a further $2$-point average of consecutive moving averages to realign the smoothed value with an actual time period.

Seasonal indices and deseasonalising

A seasonal index measures how a particular season compares with the average season. An index of $1.20$ means that season runs $20$ percent above the yearly average; an index of $0.85$ means $15$ percent below. The seasonal indices for one full cycle always add to the number of seasons (so four quarterly indices sum to $4$).

To deseasonalise a value, divide by its seasonal index:

To reseasonalise (turn a trend forecast back into an actual forecast), multiply by the seasonal index.

Fitting a trend line and forecasting

Once the series is deseasonalised, fit a least-squares trend line of deseasonalised value against the time period number $t$. To forecast a future actual value:

1. Substitute the future $t$ into the trend line to get a deseasonalised forecast.
2. Multiply by that season's seasonal index to reseasonalise.

This produces a forecast that carries both the underlying trend and the expected seasonal swing.